After enough stem cells are collected, the cells will be treated in the laboratory, using Gleevec (imatinib mesylate) followed by ex vivo culture/purging. Culture-purging is a method that takes advantage of the fact that CML cells die when kept in laboratory culture in a shorter time period than normal blood cells. If the cells are kept alive long enough, surviving cells will be less contaminated by CML cells. Imatinib mesylate will provide a chemical way of killing the CML cells, since they are more toxic to CML cells than they are to normal cells. The culture-purging will also use three medications called "growth factors". The medications being used are stem cell factor, G-CSF, and thrombopoietin.